for pinhole SPECT reconstruction
Pablo Aguiara)
Fundación Ramón Domínguez, Medicina Nuclear, CHUS, Spain and Grupo de Imaxe Molecular, IDIS, Santiago de Compostela 15706, Spain
Francisco Pino
Unitat de Biofísica, Facultat de Medicina, Universitat de Barcelona, Spain and Servei de Física Médica i Protecció Radiológica, Institut Catalá d’Oncologia, Barcelona 08036, Spain
Jesús Silva-Rodríguez
Fundación Ramón Domínguez, Medicina Nuclear, CHUS, Santiago de Compostela 15706, Spain Javier Pavía
Servei de Medicina Nuclear, Hospital Clínic, Barcelona, Spain; Institut d’Investigacions Biomèdiques August Pí i Sunyer (IDIBAPS), Spain; and CIBER en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Barcelona 08036, Spain
Doménec Ros
Unitat de Biofísica, Facultat de Medicina, Casanova 143, Spain; Institut d’Investigacions Biomèdiques August Pí i Sunyer (IDIBAPS), Spain; and CIBER en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Barcelona 08036, Spain
Álvaro Ruibal
Servicio Medicina Nuclear, CHUS, Spain; Grupo de Imaxe Molecular, Facultade de Medicina (USC), IDIS, Santiago de Compostela 15706, Spain; and Fundación Tejerina, Madrid, Spain
Ziad El Bitar
Université de Strasbourg, IPHC, 23 rue du Loess 67037 Strasbourg, France and CNRS, UMR7178, Strasbourg 67037, France
(Received 11 October 2013; revised 24 January 2014; accepted for publication 9 February 2014; published 3 March 2014)
Purpose: To assess the performance of two approaches to the system response matrix (SRM)
calcu-lation in pinhole single photon emission computed tomography (SPECT) reconstruction.
Methods: Evaluation was performed using experimental data from a low magnification pinhole
SPECT system that consisted of a rotating flat detector with a monolithic scintillator crystal. The SRM was computed following two approaches, which were based on Monte Carlo simulations (MC-SRM) and analytical techniques in combination with an experimental characterization (AE-(MC-SRM). The spatial response of the system, obtained by using the two approaches, was compared with exper-imental data. The effect of the MC-SRM and AE-SRM approaches on the reconstructed image was assessed in terms of image contrast, signal-to-noise ratio, image quality, and spatial resolution. To this end, acquisitions were carried out using a hot cylinder phantom (consisting of five fillable rods with diameters of 5, 4, 3, 2, and 1 mm and a uniform cylindrical chamber) and a custom-made Derenzo phantom, with center-to-center distances between adjacent rods of 1.5, 2.0, and 3.0 mm.
Results: Good agreement was found for the spatial response of the system between measured data
and results derived from MC-SRM and AE-SRM. Only minor differences for point sources at dis-tances smaller than the radius of rotation and large incidence angles were found. Assessment of the effect on the reconstructed image showed a similar contrast for both approaches, with values higher than 0.9 for rod diameters greater than 1 mm and higher than 0.8 for rod diameter of 1 mm. The comparison in terms of image quality showed that all rods in the different sections of a custom-made Derenzo phantom could be distinguished. The spatial resolution (FWHM) was 0.7 mm at iteration 100 using both approaches. The SNR was lower for reconstructed images using MC-SRM than for those reconstructed using AE-SRM, indicating that AE-SRM deals better with the projection noise than MC-SRM.
Conclusions: The authors’ findings show that both approaches provide good solutions to the
problem of calculating the SRM in pinhole SPECT reconstruction. The AE-SRM was faster to create and handle the projection noise better than MC-SRM. Nevertheless, the AE-SRM required a tedious experimental characterization of the intrinsic detector response. Creation of the MC-SRM required longer computation time and handled the projection noise worse than the AE-MC-SRM.
Nevertheless, the MC-SRM inherently incorporates extensive modeling of the system and there-fore experimental characterization was not required. © 2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1118/1.4866380]
Key words: pinhole SPECT, iterative reconstruction, ray-tracing techniques, Monte Carlo simulation, system response matrix
1. INTRODUCTION
Molecular imaging techniques play a valuable role in preclin-ical areas such as drug development, gene expression, and nanoparticle-based cell therapy monitoring.1–6 In particular, single photon emission computed tomography (SPECT) has become an essential tool in this field7 thanks to its ability to provide images of peptides, antibodies, and hormones la-beled with Technetium (Tc-99m) and other radio-isotopes. The relatively slow diffusion of these molecules allows the imaging of processes such as cell division, infection, and in-flammation. Furthermore, pinhole SPECT provides high reso-lution images thanks to the use of pinhole instead of parallel-hole collimators. This, enables us to achieve submillimeter spatial resolution when the object is positioned close to the pinhole.8–11This advantage comes at the expense of a reduced field of view which restricts the clinical use of these imaging systems.
On the other hand, tomographic reconstruction methods play a vital role in the quality of the images, and algorithms that include accurate description of the acquisition process can lead to high spatial resolution images. Iterative statisti-cal reconstruction algorithms have become standard proce-dures in SPECT, offering the possibility of improved image quality with respect to analytical methods.12–14 The most common statistical reconstruction method is the maximum likelihood expectation maximization (ML-EM),15 and its accelerated version, the ordered subsets expectation maxi-mization (OS-EM).16 The key element of this statistical re-construction algorithm is the system response matrix (SRM), which models the relationship between the image and the projection space. In recent years, different methods to ob-tain the SRM have been reported, those that model the SRM analytically, with or without experimental character-ization, and those that make use of Monte Carlo (MC) simulation.
In the analytical approach, the elements of the SRM are calculated using geometrical ray-tracers17–19 so that only the effects of the geometrical component are considered. The ge-ometrical contribution computes detected photons by consid-ering photon propagation in a rectilinear path as they travel in a homogeneous medium and through the point-aperture of the pinhole collimator. In order to include other effects such as intrinsic detector response, finite aperture, and sep-tal penetration, the SRM has to be factorized into several submatrices,20 each of which is related to a relevant as-pect of the image formation.21,22 The intrinsic detector re-sponse contribution computes detected photons by consider-ing photon interactions within the detector. The finite aperture
effect takes into account that the pinhole collimator is actually a finite circular aperture. The septal penetration contribution computes detected photons coming from the pinhole collima-tor edges. All this requires a laborious characterization of the system through experimental measurements, so that the sys-tem response is incorporated into the reconstruction process using different point spread function (PSF) models.23,24
When modeling is carried out using MC simulations, the resulting elements of the SRM include a full system response, thus presenting an alternative to factorization and experimen-tal characterization. To date, the robustness of these tech-niques for SPECT has already been demonstrated for both parallel25–28 and pinhole collimators,29,30 but implementa-tion involves a challenging computaimplementa-tional task since pinhole SPECT simulations are time consuming. Because of this, sev-eral variance reduction methods based on forced detection techniques have been put forward.31–33 These techniques are based on forcing each emitted photon to a circular area in the pinhole aperture, which considerably increase the number of detected events.
In summary, analytical computation of the SRM is fast but requires an experimental characterization of the system to in-clude other effects in addition to the geometrical component. On the other hand, computation of the SRM by using MC sim-ulations allows a full characterization without experimental measurements, but it is very time consuming. Nowadays, al-though it is well known that accurate modeling of the SRM is crucial in statistical reconstruction, it is still unclear which is the most suitable approach for each application. In a previous work,34we compared different analytical techniques and MC computation of the SRM for 3D PET reconstruction. Our find-ings showed that MC computation of the SRM yields small improvements in terms of contrast and spatial resolution with respect to the analytical approach.
The aim of the present work is to assess the performance of different approaches to the SRM calculation in low magnifica-tion pinhole SPECT reconstrucmagnifica-tion. The SRM was computed in line with two approaches, based on (1) Monte Carlo simu-lation (MC-SRM) and on (2) analytical techniques in combi-nation with an experimental characterization (AE-SRM).
2. MATERIALS AND METHODS 2.A. Pinhole SPECT scanner
Our work focuses on a pinhole SPECT system35 consist-ing of a rotatconsist-ing flat detector with a monolithic scintillator crystal (Cesium Iodine, CsI) of 50 × 50 × 4 mm3coupled to a Hamamatsu H8500 position sensitive photomultiplier (PS-PMT) and 1 mm diameter tungsten pinhole with a knife-edge
90◦
aperture.36 The gamma camera is attached to a variable radius system so that sensitivity and spatial resolution can be optimized by adjusting the radius of rotation to the size of the object.
2.B. System response matrix
As stated above, the key element of any statistical recon-struction algorithm is the SRM. Each element aijof this
ma-trix corresponds to the probability that a photon emission in voxel j results in the detection of an event by detector ele-ment i. Therefore, if qj is the original radioactivity
distribu-tion in voxel j and niis the statistical noise in the projection,
the measured projection pican be described by the following
equation: pi = J j =1 aijqj +ni, (1)
where i ∈ [1, I], j ∈ [1, J], and I and J are the total num-ber of projection bins and the total numnum-ber of image voxels, respectively.
2.C. Monte Carlo SRM
MC simulations were used to obtain an accurate descrip-tion of all the phenomena involved in the SPECT imaging process, which relates the activity distribution to the photons collected by the detector. The complete SRM was obtained without factorization:
r
GEANT4 simulations: The complete system response was obtained by using GEANT4 (v9.4) simulations37 of a cylinder equal in size to the field of view (FOV) and filled with a Tc-99m solution of uniform activity concentration.25,29Simulation took into account most of the physical phenomena involved in radiation-matter in-teractions, including the gamma emission, pinhole col-limation, and gamma detection. Thus, photoelectric ef-fect, and Compton and Rayleigh scatter were modeled for gamma rays whereas multiple scattering and ioniza-tion were modeled for electrons. Characteristic x rays of tungsten were not included because their contribution to the energy window centered on 140 keV is negligible. To overcome the problem of large computing time, par-allel simulations were performed on a computing grid composed of 1000 computing elements. The intrinsic re-sponse of the detector was modeled by considering two contributions: photon interactions within the detector and optical transport. The contribution of photon inter-actions was considered by tracking the photons within the detector, recording the Compton interactions for each photon in order to compute the energy-weighted centroid and total energy deposited. The tracking for op-tical photons was modeled by applying a Gaussian blur-ring of 1.6 mm FWHM.r
Variance reduction techniques: Methods for variance reduction were implemented into GEANT4 based on forced detection, so as to obtain almost noise-freesimu-lations within a reasonable time. Thus, accelerated MC simulations38 were carried out by forcing the direction of photons toward a circular area centered on the pin-hole collimator. The detected photons were weighted to compensate for the fact that emission was constrained within a limited solid angle. The number of simulated histories was 2 × 104 (realizations) in 60 projections and 106photons per projection, resulting in 12 × 1011 total simulated histories and finally 108detected events.
2.D. Analytical and experimental SRM
Computation of the different elements of the SRM can be a very tedious task, due to the large size of the matrix. To facilitate the computation, the complete matrix was divided into several submatrices,20each related to a relevant aspect in the image formation process:
r
Geometrical component: This was obtained using a fast Siddon ray-tracer.17 Each element of the matrix was computed as the length of intersection between lines (bin-voxel) and individual voxels.r
Intrinsic detector response: An experimental character-ization of the intrinsic spatial detector response was car-ried out through measurements using a pencil beam. This beam was created by collimating a source using a lead cylinder 30 cm long and 2 mm thick, with its ends closed by two circular lead covers, 4 mm thick. The source was placed on the inner side of one of the covers, while a hole of 0.5 mm in diameter was drilled in the opposite cover. A set of intrinsic detector responses was obtained for different angles of incidence and stored in LUTs according to incidence angles (from 0◦to 46◦
in steps of 2◦
) and polar angles (from 0◦
to 360◦
in steps of 2◦
).
r
Finite aperture and septal penetration: Analytical mod-els of finite aperture and septal penetration were also included. To this end, an equivalent aperture diameter yielding a geometric resolution equal to the total resolu-tion of the real pinhole was considered.39 Scatter in the collimator was not considered.Finally, each element of the SRM was generated as the convolution of a projected voxel (including the geometrical component, finite aperture and septal penetration) with the intrinsic detector response corresponding to the incidence and polar angle between the ray and detector surface (from LUTs).
2.E. Image reconstruction
Once the SRM was obtained, the maximum likeli-hood expectation maximization algorithm (ML-EM),15 with its accelerated version, the ordered subsets expectation maximization,16 was used for the reconstruction. The image is obtained from comparisons between estimated and mea-sured projection data, in such a way that each element qjof
following equation: qjk+1,n=q k j sn j In i=1aijppˆii. (2)
An iteration of algorithm is thus defined as a single pass through all the subsets, where aijis an element of the SRM; i
∈[1, In] is the projection bin; In is the number of projection bins in the subset n (n ∈ [1, SB], SB is the number of subsets); j ∈[1, J] is the image voxel, being J the number of image voxels; piis the measured projection data; ˆpi is the estimated
projection data which is obtained after forward projection of the estimated image at iteration k as
ˆ pi ≡ J j =1 aijqjk. (3) sn
j is an element of the sensitivity image s, which represents
the probability that a photon emission occurring in voxel j is detected. It can be obtained by adding all projection bin values of the subset n of the SRM in accordance with
sjn≡
In
i=1
aij. (4)
2.F. Mechanical misalignments and calibration
The mechanical misalignments and system response mod-eling are closely related topics in pinhole SPECT imaging, and a reconstruction method implemented without the ability to correct for geometric misalignments is of little interest in practical molecular imaging studies. That is why the recon-struction of pinhole SPECT data requires a correct descrip-tion of the acquisidescrip-tion geometry and this informadescrip-tion needs to be included in both the analytical calculations and MC simulations.
2.F.1. Definition of mechanical misalignments
The object is defined in the xyz system, with the z axis along the rotation axis (Fig.1) so that for a given rotation an-gle θ (anan-gle between the detector surface and the x axis), a point (x, y) of the activity distribution A(x, y, z) is projected through the pinhole with d diameter onto the point (u, v) of the detector. The origin of the uv system is perpendicular to the projection of the origin of the xyz system on the detector. To consider mechanical misalignments, the geometry of the acquisition can be completely described using seven calibra-tion parameters.40 The radius (r), is the distance between the center of the hole and the center of rotation. The focal length (f) is the distance between the center of the hole and the de-tector. The mechanical offset (m) is the distance between the rotation axis and the pinhole projection ray orthogonal to the detector. The tilt angle is the angle between the detector and the rotation axis. The t wist angle is the angle describing a ro-tation of the detector around an axis parallel to the central ray. The electrical shifts (euand ev) account for a collective
trans-FIG. 1. Rotating-camera geometry, where z axis is the rotation axis of the system, and A(x, y, z) is the activity distribution defined in the xyz system and projected into the uv system through the pinhole (d diameter).
lation of the projection image caused by a drift of the detector hardware.
2.F.2. Calibration acquisition
The seven calibration parameters (r, f, m, eu, ev, and tilt and
t wist angles) that describe the geometry of the acquisition can be obtained from a 60 projections acquisition of a three-point Tc-99m phantom.40To this end, we minimize a penalty function defined as the sum of quadratic differences of the coordinates of the projected centers of the point sources and the experimentally acquired centers.
2.G. MC-SRM versus AE-SRM 2.G.1. System PSFs
A comparison of the different spatially variant point spread functions (PSF) incorporated into the reconstruction was car-ried out. To this end, different projected point sources were obtained using the projectors estimated from MC-SRM and AE-SRM. The mechanical misalignments obtained from the calibration acquisition were included into both SRM models. Furthermore, scans of different point sources were also car-ried out, yielding profiles of the projected sources at different distances and angles from the pinhole collimator. The pro-files corresponded to the projected points placed at distances greater (27.2, 29.1, and 26.4 mm) and smaller (17.9, 16.6, and 18.1 mm) than the radius of rotation, and for three different incidence angles (4◦, 13◦and 20◦).
2.G.2. Contrast and signal-to-noise ratio (SNR)
Contrast and SNR were evaluated using a hot cylin-der phantom made of polymethylmethacrylate with internal
FIG. 2. PSFs for point sources placed at distances greater (27.2, 29.1, and 26.4 mm) and smaller (17.9, 16.6, and 18.1 mm) than the radius of rotation, and for three incidence angles (4◦, 13◦, and 20◦). The transverse profiles were obtained through projected points using MC-SRM and AE-SRM. Experimental profiles
are also shown for comparison.
dimensions of 50 mm in length and 30 mm in diameter. The phantom was composed of a fillable cylindrical chamber 30 mm in diameter and 30 mm long (uniform section). The remaining length (20 mm) of phantom was solid with five fillable rods placed at 7 mm from the center with diameters of 5, 4, 3, 2, and 1 mm (rod section). A 150 min scan time study was carried out with rod and uniform sections filled with 37 MBq/ml (1 mCi/ml) of a Tc-99m solution.
The image contrast was defined as contrast =q¯roi−q¯bg
¯
qroi+q¯bg
, (5)
where ¯qroi is the mean value in a region-of-interest (ROI)
drawn within the cylinders. These ROIs had volumes of 3.1, 12.6, 28.3, 50.3, and 78.5 mm3, respectively, and covered a
length of 4 mm; ¯qbgis the mean value of the background. The
volume of the background ROI was 113.1 mm3(6 mm
diame-ter) and also covered a length of 4 mm. In an ideal reconstruc-tion, contrast should be 1.
The uniform region of the hot cylinder phantom was also used to characterize image noise. Uniform slices were used to obtain the SNR ratio as the quotient between the mean value to the standard deviation, obtained in a transaxial central ROI, 22.5 mm in diameter. The diameter was chosen to represent 75% of the phantom diameter (30 mm).
2.G.3. Derenzo phantom
A 60 min scan time study of a custom-made Derenzo phan-tom was carried out. The hot rods had an inner diameter of
0.3 mm and the center-to-center distances between adjacent rods were 1.5, 2.0, and 3.0 mm in each of the three sections of the phantom. Acquisition parameters were 60 views (an-gle step: 6◦) and 150 × 150 projection sampling (bin size:
FIG. 3. Reconstructed images and profiles from the hot cylinder phantom by using MC-SRM and AE-SRM: reconstructed images of the hot rod section by using MC-SRM (a), AE-SRM (b), and circular profiles (c).
FIG. 4. Contrast versus number of iterations for the different cylinder diameters (a)–(e), obtained from reconstructed images by using MC-SRM and AE-SRM. SNR versus number of iterations from reconstructed images by using MC-SRM and AE-SRM (f).
0.26 mm). Reconstructed images of the Derenzo phantom were obtained using 5 subsets, 400 iterations, and a voxel size of 0.2 × 0.2 × 0.8 mm (an iteration of this algorithm was de-fined as a single pass through all the subsets).
2.G.4. Spatial resolution
The previous acquisition was also used to estimate the spa-tial resolution of the system. Thus, the rods of the section with center-to-center distances between adjacent rods of 3.0 mm (with rods sufficiently apart) were used to obtain the spatial resolution as the FWHM (mm) of the fitted Gaussian. These rods were placed at distances from the center FOV of 2.3, 3.8, 5.1, and 5.2 mm.
2.G.5. Matrix file size and computing time
Aimed at evaluating MC-SRM and AE-SRM in relation to their computational requirements, both approaches were com-pared in terms of file sizes and computing time.
3. RESULTS 3.A. System PSFs
Figure 2 shows the transverse profiles obtained through the projected points using MC-SRM and AE-SRM models. The transverse profiles from experimental measurements are also shown for comparison. First, our comparison indicates
that both SRM models are in good agreement with measured data, thereby demonstrating the accuracy of both models of system PSF. Nevertheless, a detailed analysis of the profiles shows some relevant issues that should be highlighted. The PSFs obtained from point sources placed at distances greater than the radius of rotation (27.2, 29.1, and 26.4 mm) show a better agreement than those obtained from point sources placed at smaller distances (17.9, 16.6, and 18.1 mm). In this case, PSFs obtained from the AE-SRM model were slightly narrower than those corresponding to experimental measurements, whereas PSFs obtained from the MC-SRM were slightly wider than those obtained experimentally. The latter is particularly important for an angle of incidence of 4◦
, which corresponds to a point at the edge of the detector. A comparison in terms of FWHM and FWTM showed no-table differences. Thus, the mean bias between MC-SRM and experimental PSFs was 0.32 mm in FWHM and 0.27 mm in FWTM. The mean bias between AE-SRM and experimental PSFs was −0.13 mm in FWHM and −0.60 mm in FWTM. These results demonstrate that PSF is slightly overestimated in terms of FWHM and FWTM when using MC-SRM and that AE-SRM causes FWTM to be significantly lower than the experimental measurements.
It can be seen that the projection for small angles of in-cidence (close to the perpendicular direction, for example, 4◦
) corresponds to a point at the edge of the detector, not at the center of the detector, as would be expected. This can be explained by mechanical misalignments. The geometrical
FIG. 5. Derenzo reconstructions using MC-SRM (left) and AE-SRM (right), for iteration 100 (a), 200 (b), and 2000 (c). Profiles through the rods of section with center-to-center distances of 1.5 mm are also shown (d).
parameters obtained from the calibration acquisition were the radius of rotation (22.5 mm), focal length (33.1 mm), mechan-ical offset (3.1 mm), tilt angle (0.75◦
), twist angle (0.69◦
), transverse (−4.5 mm), and axial shift (2.3 mm). It is notewor-thy that an excellent agreement exists in the position of each projected point source (measured, MC-SRM and AE-SRM), demonstrating the correct implementation of the different cal-ibration parameters.
3.B. Contrast and signal-to-noise-ratio
Figure3shows transaxial slices and profiles of hot rod sec-tion reconstrucsec-tions using MC-SRM and AE-SRM. The max-imum number of iterations for both cases was 50. The use of a higher number of iterations resulted in ring effect artifacts without increasing the contrast. A visual comparison of the re-constructed images in Fig.3shows a similar image quality for both models under investigation. Thus, both models allow us to see clearly all rods with diameter exceeding than 1 mm but not so clearly the 1 mm diameter rod. The profiles also show
an excellent agreement between the MC-SRM and AE-SRM reconstructions.
Figure4shows a quantitative comparison of the previous images in terms of contrast and SNR. As expected, contrast increases with the number of iterations and with the rod di-ameter. A similar quantitative performance can be observed for both models, showing similar contrast values for all rods after 50 iterations. On the whole, very high contrast values (>0.9) were obtained for rod diameters greater than 1 mm and relatively high contrast values (>0.8) for the rod diam-eter of 1 mm. Figure4shows that convergence for MC-SRM is slightly slower, especially for the rod diameter of 1 mm. Figure4also provides a comparison between SNR, estimated from the uniform section of the images reconstructed using MC-SRM and AE-SRM approaches. We can see that for the same number of iterations, the images reconstructed using MC-SRM exhibit higher noise.
3.C. Derenzo phantom
Figure5shows reconstructed images (iterations 100, 200, and 2000) from the custom-made Derenzo phantom acquisi-tion using MC-SRM and AE-SRM. The rods in secacquisi-tions with center-to-center distances of 2.0 and 3.0 mm can be distin-guished at iterations 100 and 200. Nevertheless, rods in the section with center-to-center distances of 1.5 mm are very fuzzy, from both MC-SRM and AE-SRM reconstructions. A proper visualization of these rods required many iterations, resulting in an increase in the statistical noise. An interest-ing difference between MC-SRM and AE-SRM was found at this point. Although all the rods could be distinguished at iteration 2000 using both models, statistical noise increased faster for MC-SRM. Hence, reconstructed images are noisier than those obtained by using AE-SRM. This lead to jagged MC-SRM profiles through three rods in this section, asym-metries in some rods and nonzero background.
3.D. Spatial resolution
The section of the custom-made Derenzo phantom with center-to-center distances of 3.0 mm was used to estimate the averaged spatial resolution values at different distances from the center of FOV, for MC-SRM and AE-SRM recon-structions. Figure6shows radial (left) and tangential (right) FWHM at different distances from the center of FOV, for it-eration 100 (a) and (b) and itit-eration 1000 (c) and (d). Spatial resolution values were improved for both approaches when the distance to the center of FOV increased, for radial and tangential FWHM. Furthermore, similar FWHM values were found for MC-SRM and AE-SRM at iteration 100. This is clearly derived from the FWHM values averaged over all dis-tances to the center of FOV, and radial and tangential direc-tions, which were 0.67 ± 0.02 and 0.68 ± 0.06, for MC-SRM and AE-MC-SRM, respectively. The spatial resolution for AE-SRM was slightly better than for MC-SRM at iteration 1000, showing FWHM values of 0.39 ± 0.02 and 0.47 ± 0.09, respectively.
FIG. 6. Radial (left) and tangential (right) spatial resolution for rods placed at different distances from the center FOV (iteration 100 and 1000). The rods are placed at distances from the center FOV of 2.3, 3.8, 5.1, and 5.2 mm.
3.E. Matrix file size and computing time
Both MC-SRM and AE-SRM were precomputed and stored in sparse format. There were major differences, how-ever between the computational requirements of the two tech-niques. MC-SRM calculations were carried out using a GRID computer cluster of about 1000 CPUs, whereas AE-SRM cal-culations were performed on a single CPU. The computing time for AE-SRM was 12 h on a single CPU and the file size containing the matrix was 11.6 Gb. The file size of the MC-SRM increased with the number of simulated photons and therefore with the computing time, converging to a file size of 12.1 GB for a computing time of 40 h on 1000 CPUs. As expected, the files for matrices from MC-SRM and AE-SRM were similar in size due to that the same sparse technique was used for compressing the matrix.
4. DISCUSSION AND CONCLUSIONS
We have assessed the effect on the final reconstructed im-ages of the two main approaches to SRM calculation for iter-ative reconstruction in pinhole SPECT. In the first approach, the SRM was computed using MC-SRM. In the second ap-proach, the SRM was computed using AE-SRM to include effects other than the geometrical component.
The comparison in terms of PSFs showed that both ap-proaches (MC-SRM and AE-SRM) are in good agreement with the measured data. Nevertheless, some minor differences for point sources placed at distances smaller than the radius of rotation and for large incidence angles were found and need to be explained. First, PSFs derived from MC-SRM ap-pear slightly wider than the measurements in terms of FWHM and FWTM, probably because of an overestimation of the optical transport effect (it seems that the PSF-shapes are similar). Second, it is clear that AE-SRM provides an ac-ceptable approach in terms of FWHM but it is a poor ap-proach for the PSF tails (FWTM) because the septal penetra-tion model fails for small source-pinhole distances and large angles of incidence. The comparison of PSFs also shows that mechanical calibration methods are essential for tomographic reconstruction. This is evidenced by the fact that when me-chanical misalignments exist, the projected points with small angles of incidence corresponded to point sources at the edge of the detector, and not at the center of the detector, as would be expected. In our case, the excellent agreement in the po-sition of projected point sources after comparing MC-SRM and AE-SRM with experimental data, demonstrates the ac-curate implementation of corrections for various mechanical misalignments. As an additional result of this work, we cur-rently have at our disposal well-validated tools for conducting
further studies on the effect of each of the calibration param-eters on image quality.
The quantitative comparison in terms of image contrast shows that a high contrast can be obtained using both ap-proaches (MC-SRM and AE-SRM), providing a very high contrast for rod diameters exceeding 1 mm and a relatively high contrast when the rod diameter is 1 mm. The compar-ison in terms of image quality and spatial resolution shows that all rods in the sections with center-to-center distances of 3.0, 2.0, and 1.5 mm could be distinguished on the basis of visual inspection and profile evaluation. Similar behavior was also found in terms of spatial resolution for both ap-proaches, which showed that spatial resolution decreased in both transverse and radial directions from the center to the periphery of the FOV. It may be hypothesized that this spa-tial resolution dependence on source position is due to the differences in convergence speed in the different regions of the FOV. This was demonstrated using images reconstructed after a very high number of iterations from free-noise pro-jections. These images showed that spatial resolution did not depend on the position in the FOV. In the experimental ac-quisitions, the number of iterations is limited because of the image noise and, as a consequence, differences in resolution can appear depending on the position of the point source in the FOV. Needless to say, the worst cases are in the center of the FOV.
Differences between the two approaches were found in im-age noise and hence for a high number of iterations. Com-parison shows that a lower SNR is obtained from MC-SRM than from AE-SRM, thereby indicating that AE-SRM deals with projection noise better than MC-SRM. Similar spatial resolution values were obtained from MC-SRM and AE-SRM (0.7 mm) at iteration 100, but improved slightly for AE-SRM (0.4 mm) at iteration 1000. This leads us to state that statistical noise limits the spatial resolution for MC-SRM, due to the fact that it is not feasible to perform more iterations at acceptable levels of noise. The differences between the two approaches regarding SNR could be attributed to the MC-SRM projector inaccuracies, which could amplify the image noise. Neverthe-less, the accurate implementation of the MC-SRM projector was demonstrated by reconstructing free-noise Derenzo pro-jections, which provided similar spatial resolution values for MC-SRM and AE-SRM, even after many iterations. It should also pointed out that the FWHM values are extremely low be-cause a capillary placed in zero-background is the most favor-able context for high iterations. Therefore, FWHM values ob-tained at iteration 1000 should not be regarded acceptable for the spatial resolution of the system. A reliable measurement of the spatial resolution of the system should be performed using a nonzero background, but this is outside the scope of the present work.
In summary, our findings show that both approaches con-stitute satisfactory solutions to the problem of calculating SRM for high performance pinhole SPECT reconstruction, showing high image quality and contrast values, and rela-tively good SNR and spatial resolution. While the AE-SRM approach is faster and handles projection noise better, it re-quires an experimental characterization of the intrinsic
detec-tor response to achieve a performance similar to that obtained using MC-SRM. The experimental characterization is a labo-rious and tedious task when trying to achieve the accuracy re-quired in this case, and an extensive experimental background may be required, thus limiting its use to experienced groups. On the other hand, the MC-SRM approach is slower, does not handle the projection noise as well and its high performance computing requirements needs a GRID computer cluster, even after the implementation of complex variance reduction meth-ods based on forced detection techniques. The implementa-tion of the AE-SRM approach requires extensive experience with a suitable programming language and the MC-SRM ap-proach can be performed using freely available packages, al-though some codes can also require knowledge of program-ming languages.
ACKNOWLEDGMENTS
This work was supported in part by Q-PET project (PI11/01806 - Fondo de Investigaciones Sanitarias del In-stituto de Salud Carlos III), the Spanish Ministry of
Sci-ence and Innovation (SAF2009-08076), and IQ-BRAIN
project (10CSA918001PR - Xunta de Galicia). P. Aguiar was awarded a “Sara Borrell” fellowship (CD09/00291) by Fondo de Investigaciones Sanitarias del Instituto de Salud Carlos III. The authors also acknowledge the support of IPHC in pro-viding computing resources on the Regional Grid Infrastruc-ture (http://www.grand-est.fr).
a)Author to whom correspondence should be addressed. Electronic mail:
pablo.aguiar.fernandez@sergas.es
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